Full-weight conjecture for weighted pebbling numbers of trees

Let $T$ be a tree, let $E(T)$ denote its edge set, and let $wp(T)$ be its weighted pebbling number. For a weighted graph, write $w_T(k)$ for the minimum total edge weight required for $T$ to achieve weighted pebbling number $k$. Full-weight conjecture. Every tree requires the full weight $|E(T)|/2$ to achieve its weighted pebbling number; equivalently,

$$w_T(wp(T))=|E(T)|/2.$$

The paper contrasts this claim with graphs such as $K_5$, where less than $|E(K_5)|/2$ can suffice, and notes that the full weight is already known to be necessary for paths, stars, and the tree considered in the preceding proposition. The conjecture asserts this necessity for all trees.